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Computational Astronomy and the Earliest Visibility of Lunar Crescent

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Computational Astronomy and the Earliest Visibility of Lunar Crescent COMPUTATIONAL ASTRONOMY AND THE EARLIEST VISIBILITY OF LUNAR CRESCENT Muhammad Shahid Qureshi In charge & Assistant Professor Institute of Space & Planetary Astrophysics University of Karachi ABSTRACT Basic techniques of Computational Astronomy are reviewed and presented as the essential tools for simulation of Lunar phenomena. The importance of accurate determination of Julian Date and the Local Sidereal Time is discussed that are essential to determine the local time of sunset and the local coordinates of any object at that time. During the 20th century, a number of authors have contributed towards the understanding of the problem of earliest sighting of crescent Moon. The work of Maunder, Schoch, Bruin and Schaefer has been crucial in the development of this understanding. More recently, the work of Yallop, Ilyas, Ahmed and Shaukat has received great recognition. The work of Ahmed and Shaukat has been based mostly on the Yallop’s Criterion. However almost all the modesl are based on the observational data of Schmidt who made observations from Athens for over 20 years during the late 19th century. In this work, a model of q-values developed by Yallop is analyzed in view of Maunders and the Indian Criteria along with the actual semi-diameter of the crescent Moon. The basic criterion is modified on the basis of data more recently collected. 1. INTRODUCTION Observations or results of experiments form the basis of all theories describing any physical phenomenon. More minute observations and sophisticated experiments provide testing grounds for these theories. Kepler[18][26] deduced his laws, describing motion of planets, based on data available in his time. On the basis of his dynamical theories Newton proved Kepler’s laws. However, Kepler’s Laws of Planetary motion are valid only in view of the Two-Body problem[4][26]. For instance, the orbits of planets that are assumed elliptical by Kepler do not remain elliptical even if a Three-Body problem is considered. Much later, the Newtonian Dynamics that may be effectively used to describe n-Body problem, itself failed to describe the variations in the orbit of Mercury. Even the Relativistic Mechanics of Einstein could not completely account for this variation. Description of these Physical theories and putting them to work to determine the position of celestial bodies on our skies are two different aspect of the same effort. Observed positions lead to theories and a theory has to be tested by observations. In the first part of this paper, tools of Computational Astronomy are presented, in view of the problem of determining the position of a planet or a satellite in its orbit and as it appears on the observable sky. In particular, the Two-Body[26] and the Three-Body[4] Problems are briefly reviewed and how they are used to determine positions of the Sun and the Moon in the sky. In the 1 second part the problem of earliest visibility/sighting of New Crescent Moon is presented. The work of a number of contributors and authors is reviewed. Yallop’s comparison[29] of the Maunder’s[13], the Indian[11] and the Bruin’s[3] methods is extended to include a q-test based on third degree polynomial fitted to data using Least Square Approximation. Moreover the possibility of better coefficients in the 2nd degree curve is considered to be closer to real observations. In the end, astronomical conditions for a number of recently recorded observations calculated on the basis of each method are presented with a critical discussion. 2. COMPUTATIONAL ASTRONOMY In this work, all the tools of Computational Astronomy are based on Newtonian Mechanics. The time argument used is based on Julian Date according to Gregorian Calendar[17]. The Julian Date is thus the time elapsed since the Noon at Greenwich on Monday, November 24 of the year - 4713. It is intended that this work shall be extended to higher degree of precision in order to develop an indigenous and independent simulation for the celestial phenomena considered in this work. Most of the tools discussed in this article and the next one are due to Smart[26], Danby[4] and notes from the Astronomical Almanac[28]. 2.1 SPHERICAL TRIGONOMETRY We observe all celestial objects moving on the “Celestial Sphere”. To determine the relative position of objects, angular separation between them and direction of one from the other we require Spherical Trigonometry, the study of triangle on a sphere. The shortest “distance” between two points on a sphere is along a “great circle arc”. A “great circle” is the intersection of a plane passing through the centre of the sphere with the sphere. Any plane that does not pass through the center of the sphere intersects with the sphere in a “small circle”. Let A, B and C be points on a sphere. AB, BC and AC are great circle arcs with “length” equal to c, a and b respectively, given in angular measures as shown in Figure 1 page 3. Thus a = ÐCOB,b = ÐAOC and c = ÐAOB , where O is the centre of the sphere. The basic formulas in spherical trigonometry are: Cos(a) = Cos(b)Cos(c) + Sin(b)Sin(c)Cos(A) (2.1) Sin(A) Sin(B) Sin(C) = = (2.2) Sin(a) Sin(b) Sin(c) Where A is the spherical angle between the sides b and c which is the angle between the planes the circular sectors AOC and AOB. Similarly the spherical angles B and C are defined. All the other formulas of spherical trigonometry can be obtained using these two formulas. First of the above two formulas, (2.1), gives the angular separation ‘a’ between points B and C on the sphere. Direction of a point say A from a point B is given by the spherical angle C. The two formulas can also be used to find the “shortest distances” between two places on Earth and the direction of one place from another, for instance direction of Qibla from any place on the Earth. 2 Figure 1 Figure 2 2.2 TERRESTRIAL COORDINATE SYSTEMS Latitudes and Longitudes give position of any point on the surface of earth. As shown in Figure 2 above, the reference for Latitude is the Equator, the great circle that passes through the centre of Earth and its plane is perpendicular to the axis of rotation of Earth. Semi great circles joining the North Pole(point of intersection of the surface of Earth with the axis of rotation in the northern hemisphere) and the South Pole are called Meridians. The Meridian Passing through Greenwich, near London, England is called the standard meridian. In figure 2, G is Greenwich and K is any other place, say Karachi. NGAS is the standard meridian and NKBS the local meridian of Karachi. If O be the centre of Earth then Latitude of Karachi is the angle ÐKOB denoted by j and the Longitude of Karachi is ÐAOB denoted by L. If the place K (j, L) is in the upper hemisphere its Latitude is between 00 North and 900 North. If the place is towards East of the standard meridian then its Longitude is between 00 East to 1800 East. The southern latitudes and the Western longitudes are defined in the same way. 2.3 CELESTIAL COORDINATE SYSTEMS There are a number of celestial coordinate systems that are essential for the understanding the computational astronomy. These are discussed briefly in the following: 2.3.1 GEOCENTRIC ECLIPTIC COORDINATES The positions of points on the celestial sphere are defined in a number of ways. If the equatorial plane of the Earth is extended up to the sky, the celestial sphere, it intersects the sky in a great circle called the Celestial Equator, as shown in Figure 3. For a person standing on the equator of Earth the celestial equator extends from due east to due west passing right above the head of the person through a point called Zenith(the highest point in the sky). For an observer in the northern hemisphere the celestial equator also extends from due east to due west but remains south of the 3 zenith. Similarly if the axis of rotation of Earth is extended to the sky it meets the celestial sphere in points P called the North Celestial Pole and the Q the South Celestial Pole. Although, it is Earth that revolves round the Sun, but relative to an observer on Earth the Sun travels around the Earth along a great circle called Ecliptic. The planes of the Ecliptic and the Celestial Equator are inclined to each other at angle e = 230.5 , called the Obliquity of the Ecliptic. The two great circles, the ecliptic and the celestial equator, intersect in two points g and g¢ , called the Vernal Equinox and the Autumnal Equinox. In its “journey” around Earth, the Sun reaches g on March 21, and g¢ on September 21. g , also known as, First Point of Aries, and the Ecliptic form the reference of the Ecliptic Coordinate system. The point K on the celestial sphere, which is at 900 from each point on the ecliptic, is called the Pole of the Ecliptic. If S is any point on the celestial sphere then the let KSH be great circle arc joining K through S the ecliptic at H. The Ecliptic Latitude denoted as b , of S is the angle ÐSOH and the Ecliptic Longitude denoted as l of S is the angle ÐgOH (where O is now the centre of Earth). Thus (l,b) are the Geocentric Ecliptic Coordinates of S. Figure 3 2.3.2 GEOCENTRIC EQUATORIAL COORDINATES For the Celestial Equatorial Coordinates the references are the Celestial Equator and the Vernal Equinox, as shown in Figure 3 above.
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